Supplemental Lecture 21 Bosonization in 1+1 Dimensions and Solving the Schwinger Model Introduction to the Foundations of Quantum Field Theory For Physics Students Part VIII Abstract We derive the duality map between fermions and bosons in 1+1 dimensions and apply this method to solve the Schwinger model, quantum electrodynamics in 1+1 dimensions. For massless interacting fermions, the solution is a free but massive neutral boson. The physics of the model is discussed from the perspective of quantum choromodynamics and other non-abelian gauge theories. The Schwinger model illustrates the physics of flux tubes, confinement, the Higgs mechanism, the chiral anomaly, chiral symmetry breaking, mass generation and theta- vacua in a simple setting. Prerequisite: This lecture continues topics started in Supplementary Lectures 19 and 20. This lecture supplements material in the textbook: Special Relativity, Electrodynamics and General Relativity: From Newton to Einstein (ISBN: 978-0-12-813720-8) by John B. Kogut. The term “textbook” in these Supplemental Lectures will refer to that work. Keywords: Bosonization, 1+1 Dimensions, Correlation Functions, Operator Product Expansions, Confinement, Higgs Mechanism, The Chiral Anomaly, Chiral Symmetry Breaking, Mass Generation, The Schwinger Model. Quantum Electrodynamics in 1+1 Dimensions, Theta Vacua. --------------------------------------------------------------------------------------------------------------------- 1 Contents 1. Introduction. Goals of This Lecture. ....................................................................................... 2 2. Fermions in 1+1 Dimensions. ................................................................................................. 3 2. Bosonization in 1+1 Dimensions. ........................................................................................... 6 3. The Fermion-Boson Dictionary. ........................................................................................... 12 4. The Solution of the Schwinger Model. ................................................................................. 14 References ..................................................................................................................................... 18 1. Introduction. Goals of This Lecture. The Schwinger model, electrodynamics in 1+1 dimensions, was introduced and solved by J. Schwinger [1]. The model was popularized as a simple model of confinement in the early days of quantum chromodynamics [2]. In fact, the model stimulated a great deal of productive thinking about non-perturbative phenomena in field theory in an era previously dominated by perturbation theory. The model provided fundamental insights into the physical origin of the chiral anomaly, mass generation, confinement and the Higgs mechanics. The problem of confinement in 3+1 dimensional quantum chromodynamics consists of the following challenges: can the theory’s colored quarks be weakly coupled at short distances, but be strongly coupled at large distances so that they cannot be isolated? From the perspective of an experimentalist, can a theory exist in which its fundamental constituents can be observed and have their quantum numbers measured in high resolution experiments, like deep inelastic scattering, and yet those fundamental constituents cannot be produced individually in the final states of those experiments? Early in the quantum chromodynamics era, these ideas were highly debated, although now they are accepted as orthodoxy, even though they have not been derived from analysis of quantum chromodynamics. This is why model field theories were very 2 important in the early days of quantum chromodynamics and the demonstration of confinement in the Schwinger model was an important step in the development of the field. In 3+1 dimensions one imagines that electric flux tubes form at large distances due to the running of the theory’s coupling constant to large values at large distances, a fact supported by numerical simulations of SU(3) Yang Mills theory in 3+1 dimensions. In 1+1 dimensions, electric flux cannot spread out and the classical model confines charge with a linear force law so confinement in the Schwinger model with massive fermions is not surprising. However, the fact that the quantized theory is a free massive scalar field even in the limit where the fermion mass goes to zero is quite striking and inspirational. The quantum nature of the Schwinger model and its non- trivial vacuum are essential here. We have already seen how the Dirac sea plays a critical role in the theory’s violation of chiral symmetry and its chiral anomaly. Here we will see that the chiral anomaly leads to the mass generation of the scalar field. The solution of the Schwinger model will be done through bosonization in order to illustrate a non-perturbative method. Actually, the physics of the model can be extracted via perturbative methods because of the simplicity of kinematics in 1+1 dimensions. The bosonization method grew out of methods originally developed in statistical mechanics to analyze and solve two dimensional spin systems [3]. The cross fertilization between statistical field theory methods and high energy physics field theory methods was essential here and has led to many joint triumphs more recently. 2. Fermions in 1+1 Dimensions. We have discussed fermions in 1+1 dimensions in the previous two lectures. In the chiral representation in 1+1 dimensions used in the previous lecture, the action reads, 2 † † 푆 = ∫ 푑 푥 (푖휓+(휕0 − 휕1)휓+ + 푖휓−(휕0 + 휕1)휓−) 2.1 Here 휓± are chiral eigenstates as discussed in Sec. 4 of lecture 20. In that lecture we wrote out 휇 ̅ 휇 휇 ̅ 휇 5 and studied the vector current 푗 = 휓훾 휓 and the axial vector current 푗퐴 = 휓훾 훾 휓. Let’s write out these fields 휓± in terms of fermion creation and annihilation operators [4]. It follows from Eq. 2.1 that the chiral fermion 휓− is right-moving. We interpret 휓− as a quantized field from here forward, following the formalism for complex scalar fields introduced 3 in lecture 14, except for the changes needed to execute bosons → fermions that will be discussed below. We use the Schrodinger picture where states have time dependence but operators do not. So, 휓− will be expanded in terms of creation and annihilation operators for 푝 > 0. Introduce (−)† (−) a creation operator 푐푝 for anti-particles and an annihilation operator 푏푝 for particles, ∞ 푑푝 ( ) ( ) 휓 (푥) = ∫ (푏 − 푒푖푝푥 + 푐 − †푒−푖푝푥) 2.2a − 0 2휋 푝 푝 And similarly for 휓+ which is left-moving, 0 푑푝 ( ) ( ) 휓 (푥) = ∫ (푏 + 푒푖푝푥 + 푐 + †푒−푖푝푥) 2.2b + −∞ 2휋 푝 푝 The creation and annihilation operators are postulated to obey anti-commutation relations, (±) (±)† (±) (±)† {푏푝 , 푏푞 } = {푐푝 , 푐푞 } = 2휋훿(푝 − 푞) 2.3 with other anti-commutators vanishing. For example, (±)† (±)† {푏푝 , 푏푞 } = 0 2.4 (±)† (±)† So 푏푝 푏푝 = 0 which insures the Pauli exclusion principle, two fermions cannot occupy the (±)† (±)† (±)† (±)† same state, and 푏푞 푏푝 = −푏푝 푏푞 which enforces the fact that when fermions are interchanged, the wave function changes sign. The vacuum is defined by the conditions that it is (±) (±) (±) (±) (±)† (±)† annihilated by 푏푝 and 푐푝 , 푏푝 |0 >= 푐푝 |0 >=0. The 푏푝 creates particles and the 푐푝 creates anti-particles. The anti-commutation relations Eq. 2.3 and 2.4 imply the basic anti-commutation relations for the field operators. For example, using Eq. 2.2a and 2.2b one computes, † {휓±(푥) , 휓±(푦) } = 훿(푥 − 푦) 2.5 with the other anti-commutators vanishing. Field theories need careful analysis at very short distances and very high energies. To avoid ambiguous expressions we modify Eq. 2.2 to read, ∞ 푑푝 ( ) ( ) 휓 (푥) = ∫ (푏 − 푒푖푝푥 + 푐 − †푒−푖푝푥)푒−푝/2Λ 2.6a − 0 2휋 푝 푝 4 and 0 푑푝 ( ) ( ) 휓 (푥) = ∫ (푏 + 푒푖푝푥 + 푐 + †푒−푖푝푥)푒−|푝|/2Λ 2.6b + −∞ 2휋 푝 푝 where Λ is a momentum cutoff which must be taken to ∞ at the end of all calculations. 1⁄Λ appears in many calculations so we define 휖 ≡ 1⁄Λ. We will need the vacuum expectation values of products of field operators [4]. For example, ∞ 푑푝 푑푞 ( ) ( ) < 휓 (푥)휓†(푦) > = ∫ < 푏 − 푏 − † > 푒푖푞푥−푖푝푦−(푝+푞)/2Λ − − 0 (2휋)2 푞 푝 ∞ 푑푝 푖 1 = ∫ 푒푖푝(푥−푦+푖휖) = 2.7 0 2휋 2휋 (푥−푦)+푖휖 Let’s understand some aspects of this result. First, note that it depends on the difference 푥 − 푦 as a consequence of translation invariance. In addition, it varies as (푥 − 푦)−1 by dimensional −1/2 † analysis: each 휓− carries the dimensions [퐿 ], so 휓−(푥)휓−(푦) should vary as an inverse length and the only candidate is 푥 − 푦. We also see that the high momentum cutoff Λ was essential to obtain a well-defined result when 푥 → 푦. This is reasonable because when 푥 → 푦, the high momentum fluctuations in the fields become especially important. If the theory were interacting, all these issues would come up again, but the dynamics would make analogous calculations more challenging! In such theories, the product of two operators at 푥 and 푦 and their singular behavior as 푥 → 푦 contains essential information concerning the theory’s high energy dynamics. Can we retrieve the basic anti-commutator Eq. 2.5 from this analysis? First, a short † calculation like the one just done, shows that < 휓−(푥)휓−(푦) > equals Eq. 2.7. Therefore, the vacuum expectation value of the commutator is, 푖 1 1 1 휖 < {휓 (푥), 휓†(푦)} > = ( + ) = 2.8 − − 2휋 (푥−푦)+푖휖 −(푥−푦)+푖휖 휋 (푥−푦)2+휖2 and the right hand side becomes 훿(푥 − 푦) as 휖 → 0, recalling a basic formula from one’s introductory course in quantum mechanics, 1 휖 lim = 훿(푥 − 푦) 휋 휖→0 (푥 − 푦)2 + 휖2 5 † † Other bits of algebra we will need later are : < 휓+(푥)휓+(푦) > = < 휓+(푥)휓+(푦) > , and 푖 1 < 휓 (푥)휓†(푦) >= − 2.9 + + 2휋 (푥−푦)−푖휖 2. Bosonization in 1+1 Dimensions. It was known since the 1960s that there are intimate relations between fermion fields and the exponentials of boson fields in 1+1 dimensions [4]. They were part of the bag of tricks specialists in model field theories had developed. Parallel developments had occurred in statistical physics. For example, the Jordan-Wigner relation had been used to rewrite versions of the two dimensional Ising model in terms of Dirac and/or Majorana fermions.